HIRTOBY OF THE THEOKY OF NUMBERS. [CHAP, vm -oved that, if p is a prime, there are exactly 0 (mod p) not having as roots k given distinct num- -l)fc. For n = fc, ^(n, &) is the number i£(n) of *th no root. The number with exactly i roots is *V(i— r) distinct matrices (3) of rank i such . a1; . . . not divisible by p. $d a function <£>(/) of a polynomial f(x) such e coefficients of f(x) are increased by integral -•et/fc^CaO, i= 1, . . ., p*, denote the polynomials act modulo p and have unity as the coefficient 2 */«(*) -S * takes those values 1, 2, . . ., pn for which f(*\x)=Q (mod p) does ve as a root one of the given incongruent numbers ab . . ., as; while, *« ~ue outer sums on the right, i, if, . . . range over the combinations of 1, . . . , s without repetitions. Zsigmondy38 had earlier given the preceding formula for the case in which oi,. . ., aa denote 0, 1,. . ., p — l. Then taking <£(/) = !, we get the number of congruences of degree n with no root (Zsigmondy36). Taking $y) =f} we see that the sum of the congruences of degree n with no root is =0 (mod p), aside from specified exceptions. Taking <£(/)= a/, where co is a pth root of unity, and n^p, we see that the system /^(z) takes each of the values 1, . . ., p — l (mod p) equally often. Zsigmondy39 proved his36'37 earlier formulas, obtained for an integral value of x the number of complete sets of residues modulo p into which fall the values of the/(^(x) not having prescribed roots, and investigated the system Bn of the least positive residues modulo p of the left members of all congruences of degree n having no root. In particular, he found how often the system Bn contains each residue, or non-residue, of a <?th power. He investigated (pp. 19-36) the number of polynomials in x which take k prescribed residues modulo p for k given values of x. "Sitzungsber. Ak. Wis9. Wien (Math.), 103, Ila, 1894, 135-144. "Monatshefte Math. Pkys., 7, 1896, 192-3. "Jahresbericht d. Deutschen Math. Verein., 4, 1894-5, 109-111. "Monatshefte Math. Phys., 8, 1897, 1-42.